Dielectronic recombination rates for ions of the magnesium sequence at low energies

J Quanr Sp~crrnsc Rodm Printed in the U.S.A.

Trans/?r Vol. 33. No. I, PP. 13-26. 1985

0022-4073/85 $3.00 + .I0 Pergamon Press Ltd.

DIELECTRONIC RECOMBINATION RATES FOR IONS OF THE MAGNESIUM SEQUENCE AT LOW ENERGIES M. P. DUBE, R. RASOANAIVO, and Y. HAHN Department of Physics, University of Connecticut, Storm, CT, 06268, U.S.A. (Received 15 July 1983) Abstract-The dielectronic recombination (DR) rate coefficient (I”~ ISexplicitly calculated for the Si, Ar, Fe and MO target ions of the Mg isoelectronic sequence with twelve electrons. Both the 3s, An # 0 and 3s, An = 0 transitions are considered in detail. An explicit LS coupling scheme was applied to all the dominant transitions of these ions. Results of aDR with different free electron temperatures are also discussed 1. INTRODUCTION

Dielectronic recombination (DR) of free electrons with partially ionized atoms is an important process in high-temperature, low-density plasmas. Precise values of recombination rate coefficient for many different ionic species are needed for the mathematical modeling of tokamak plasmas,‘** and it is also found to be important in the solar corona.3’4 As a continuation of previous theoretical studies5*6 of DR rates and cross sections, we present in this paper the DR rates aDR for the Mg sequence at low energies. In Sec. 2, the relevant theory and calculational techniques are reviewed. As a preliminary to a more detailed calculation in Sec. 3, the results of the aDR using the angular momentum averaged (AMA) approximation is given for all the important intermediate states of Ar6’. From this set, a subset of dominant states is selected and the DR rate coefficients are recalculated for the subset using the full LS coupling scheme. For the ions Si*+, Ar6’, Fe14+, and Mo3’+, the calculation is separated into two types: 3.5 An # 0 and 3s, An = 0. These 3s excitations are the most important ones at low energies and will be considered in this paper. The other excitations involving the 2p, 2.5 and 1s electrons are more important at higher energies and these will be considered later in a separate study. The configuration mixing and intermediate coupling effects will also be examined there. In Sec. 4, the aDR with different temperatures are reported. Also included is a discussion on different angular momentum recoupling procedures, along with the results of several sample calculations on rate coefficients based on two different recoupling schemes. 2. PRELIMINARY

DISCUSSIONS

Dielectronic recombination (DR) is a two-step process in which a free electron is captured by a positive ion in its initial state i and gives rise to any one of a number of narrowly resonant, doubly excited intermediate states d. This intermediate state then emits a photon and the resulting ion becomes stable in the final state f: The process is described as AZ + e- - (A’-‘)** -

(A’-‘)* + y.

(1)

Here, Z is the initial degree of ionization of the atom A. The cross section of this process aDR is used to define the DR rate coefficient as

aDR= (vcaDR)T, where v.‘,is the free electron velocity and ( ) T indicates a (Maxwell-Boltzmann) thermal average. For a given initial state i, aDR (in cm3/sec) at a temperature ksT is given in the isolated resonance approximation by 13

14 aDR(i)

M. P. DUBE et al. =

2

C

aDR(i,

1, -

d)

=

(47rRy/ksT)3’2a03

I, d

2 2 exp(-e,(d)/ksT)I/,( I< d

i, 1, -

d)w(d), (3)

where a0 = 0.529 X lo-* cm. The summation index d specifies all of the relevant quantum numbers associated with a given intermediate state. 1, and e,(d) are, respectively, the angular momentum and energy of the free continuum electron at resonance with the state d. The quantity V,(i, 1, - d) in set-‘, is the collisional excitation-capture probability and is related to the Auger probability by detailed balance as

A,(d -

i, 1,) = gd V,(i, 1, 2g,

d).

The statistical weights of the intermediate and initial states are denoted by gd and g,, respectively; w(d) is the fluorescence yield and is given by (neglecting possible cascade effects) o(d) = r,/V’, where the radiative

+ r,),

and Auger decay widths are defined

r,(d) = C AAd -+A

(5) as

I’,(d) = 2 A,(d -

i).

I

./

(6)

In Eq. (6) the A,(d -f) and A,(d - i) are the radiative and Auger decay rates for the particular transitions, in units of see- ‘. Explicit formulae have already been given in Ref. 5. Bound state wave functions used in the evaluation of the A, and A, were generated by a nonrelativistic, single-configuration, self-consistent Hartree-Fock code.’ Continuum electron wave functions needed for A, were calculated numerically in the Hartree-Fock potential field of the target ion, with the non-local exchange effect included explicitly. The entire calculations were partitioned into excitation classes and these classes are treated separately. The excitation classes are Is, 2s, 2p, 3s, An # 0, and 3s, An = 0. Only the 3s, An # 0 and 3s, An = 0 excitation classes which are dominant at low energies are considered in this paper. 3.

CALCULATIONS

AND

RESULTS

OF

DR

RATE

COEFFICIENTS

A. 3% dn = 0 Excitations All the target ions considered here (Si’+, Ar6+, Fe14+, and Mo3’+) contain twelve electrons and thus belong to the Mg isoelectronic sequence. Their initial configuration is taken to be ( ls22s22p6)3s2. The configuration mixing of the 3s2 with 3p2 and 3s3d states is neglected in the present study, although its effect may be important. This simplification is consistent with the distorted-wave Born approximation employed here. The core electrons ls22s22p6 do not change much as the outer-shell electrons are excited, and so we simply denote them collectively by (x) below. For the 33, An = 0 excitations, the 3s electron can be excited into either the 3p or 3d orbital, and this implies that the intermediate states are of the forms (x)3s3pnl and (x)3s3dnl, where nl designates the orbital occupied by the captured electron. The principal quantum number n varies from some minimum value no to infinity. The angular momentum quantum number 1 varies of course from 0 to n - 1 for each fixed n, but, in a typical series of calculations, 1 is held fixed at some value and n varied from no to infinity. The contribution from each 1 is then summed over, but the high l(1 Z 10) effect is found to be negligible. For the (x)3s3pnl intermediate states, only one initial state is possible, i, = (x)3?, whereas for 3s3dnl there are two initial states to which they can decay, i.e., i, and i2 = (x)3s3p. The next section discusses the 3s, An # 0 transitions, and they can have three initial states, i, , i2, and i3 = (x)3s3d.

Dielectronic

recombination

rates for ions of the magnesium

sequence

15

Table I. Values of energy gaps AE for the ions Si’+, API, Fe14+, and MO’“‘. Values for the Si and Ar ions are obtained from Moore’s tables;” values for the Fe and MO ions are obtained by extrapolation linear in Z. The energy is given in units of Ry.

IOIl

AE(3s-3~)

AE(3s-3d) Singlet

Singlet

Si2+

dE(3p-3d) Singlet

AE(3p-3d) Triplet

0.755

1.31

0.550

0.634

1.555

3.374

1.83

1 .g1

Fe’4+

3.155

7.555

--__-

--__-

!lo30+

6.350

--_--

--__-

--__-

Ar

6+

The minimum value no corresponds to the capture of a very low energy free electron, and can be determined by the energy conservation as edno)

=

AE - e(nol) = 0.

(7)

In Eq. (7) AE is the 3s-3p or 3s-3d excitation energy, and e(n& is the energy of the captured electron in the (no,) orbital. The energy e(n& can be approximated by the single particle energy e(n04

(8)

-Z*/(n,*)*,

-

defect p for the state no1 and where Z is the where n,* = no - p with the quantum effective charge that the electron in this state sees. (Generally, p is negligible for I > 2.) Therefore, from (8) we have no -

Z/a.

(9)

The 3s-3p and 3s-3d excitation energies are given in Table 1 and a list of the minimum values of no for the four ions are given in Table 2. It can easily be seen from Table 2 that the excitation of a 3s electron to the 3p state is accompanied by the capture of an incident electron to a high Rydberg state (HRS), and this phenomenon of capture to HRS becomes more prominent as the nuclear charge of the ion increases. For example, the Mo3’+ ion allows the lowest intermediate state (d) of 3~3~121, with no = 12. The subject of HRS in DR has been dealt with in detail in a previous paper.’ for several low In a typical series of calculations, the aDR are explicitly calculated values of n starting from the minimum value no. The necessary Auger decay rates A, and radiative decay rates A, are calculated numerically. Next, to extend the calculation to larger n, we can use either the quantum defect theory and extrapolate down from the continuum or the scaling properties of the A’s and extrapolate up to high n; a direct extrapolation of the DR rates themselves can sometimes lead to a serious error. It can

Table 2. Minimum

values no of the principal quantum number Mo3’+, as evaluated from LU? of Table

IOn

Si2+ Ar

6+

3s3dnO/j

353pnJ

no

=

n, for the ions Si”.

I and Eq. (8).

4

n ,=!

5

4

Fe’4+

8

6

M03OC

12

9

A?,

Fe“‘+. and

M. P. DUBE et (11.

16

easily be shown that the Auger decay rates A, and the Auger width I’0 will decrease as l/n3 when n increases and is sufficiently large. On the other hand, rr contains a part which behaves as d/n3 and another part which is independent of n. That is, I’, - b/n3,

A, - a/n”,

r, - c + d/n’,

c + dJn’

o(d) -

c + (b + d)/n’ ’

(10)

where a, b, c, and d are constants. The constant c corresponds to the radiative decay rates for 3p to 3s, or 3d to 3p, in the electric dipole approximation, and thus are independent of n. Equation (10) implies that as n increases, w(d) approaches one and aDR scales like l/n3. Therefore, the total sum of all the aDR can easily be obtained as n varies from no to infinity. For n b no and I >, 5, the exchange part of the amplitude in A, becomes very much smaller than the direct part and can be neglected. The wave functions for the high Rydberg state used in these calculations were approximated by pure hydrogenic values. It is also known’ that not all of the final states to which the intermediate states d decay radiatively are Auger stable against electron emission. To incorporate this cascading effect, the fluorescence yield is modified to

Table 3. Sample values of A, for the ions Si”, A?‘, Fe14’ and MO”“. In the table the three initial state are denoted by i, = (x)3?, iz = (x)3s3p, and i, = (x)3s3d. The blanks denoted by - - - (I - - - imply that the minimum value of no for a particular ion is greater than that given in Table 2: the blanks denoted by - - -h- - - imply that there are no final states which are stable: - - -c- - - imply that the values were found to be small and therefore were not considered.

-

-

I1 al

s ab

7 4 4

2 2 4 4

0 1 0 1

2.17+11 5.85+12 4.41+13

2 2 4 4

2 2 4 4

0 1 0 1

2 2 4 4

2 2 4 4

2 2 4 4

L

(xl3s3p4f

(x)3s?p6f

7

Si

Ai-

___a-__

MO

Fe

___a___

-__3__-

_--a_-_

-__a___

___a___

___a___

_--a___

__-a__-

9.93+13

_-_a___

_--a_-_

_--a___

1.49+11 2.80+12 1.89+13 4.02+13

2.86+13 7.14+13 2.38+14 4.:6+14

_--a__-

_--a-__

___a__-

_--a___

0 1 0 1

7.81+10 1.31+12 8.58+12 1.80+13

7.76+12 1.78+13

2 2 4 4

0 1 0 1

i

;

;

_--a-_-

___3_-._

___a__-

_-_a___

_-_a__-

4.73+13 7.74+13

2.36+13 4.34+13 4.30+13 5.45+13

___c-__ ___c___ --_c_-_ ___c___

___c_-_ ___c-__ -__c___ _-_c___

___c_-___c_-_ ___c___ ___c_-_

9.72+12 1.95+13 5.15+12 4.62+12

0 1

--_b_-_ -__b_--

2.79+13 %.51+13

7.07+13 3.73+13

1.01+14 4.45+13

:

0 1

_--b_-_--b__-

1.65+13 1.87+13

3.39+13 2.87+1X

3.98~13 3.72+13

3 3

:

0 1

--_b_-_-_b--_

9.72~12 1.11+13

1.79+13 1.78+13

1.98+13 ?.36+11

2 2 4 4

2 2 4 4

0 1 0 1

-__b___ ___b___ ___b___ _--b_--

1.14+13 1.75+12 3.36+13 2.40112

3.24+11 2.68+10 4.34+12 8.72+11

___a___ ___3___ _-_a___

(X)3S’iS5f

2 2 4 4

2 2 4 4

0 1 0 1

___b___ -__b___ -__b___ ---b---

9.38+12 2.91+11 1.82+13 3.11+12

7.06+09 l.ll+ll 5.78+11 4.92+09

2.50+12 6.01+11 2.47+12 1.00+12

(Xl3S4S5f

1

1

: 3 5 5

: 3

0 1 0 1 0 1

-__b___ ___b___ ---b--_ ---b__-__b--_ _-_b--_

5.32+13 1.46+13 1.30+12 3.16+11 4.30+12 9.81+10

2.02+13 5.30+12 1.21+13 4.88+10 1.14+12 3.84+10

8.73+12 1.89+12 1.90+13 1.77+10 3.32+09 3.01+11

(X)3S3PEf

r

-

-

55

___3___

_-_a____-a___

---a___

-

Dielectronic

w(d) = &

recombination

rates for ions of the magnesium

C iA,(d 4 f) + C A$(;)

f

17

sequence

‘) [A@’ + f)

* * *I} 3

(11)

d

where I’(d) = I’,(d) + I’,(d). The states labeled fare stable against electron emission, whereas the states labeled d are Auger unstable, i.e., the states labeled d can decay radiatively and/or by electron emission. The cascade corrections decrease the value of w(d) and keep it from approaching one. This in turn reduces the values of aDR. Sample values of A, and A, are presented in Tables 3 and 4. In these tables, there are several reasons why particular values for A, and A, may be missing; the blanks denoted by-_-a-__ imply that the minimum value of no for a particular ion is greater than that given in the table; the blanks denoted by - - - b - - - imply that there are no final states which are stable against electron emission; the blanks denoted by - - - c - - imply that the values were found to be small and therefore were not considered. In transitions in the LS Tables 5-8, the rate coefficients aDR are given for the dominant coupling scheme; these are the 3s, An = 0 excitations for the ions Si2+, Ar6+, Fe14+, and Mo3’+. When the effect of the spectator electron is included, these results are consistent with the result for ions of the Na isoelectronic sequence.6 B. 3s, An # 0 Excitations For the 3s, An # 0 excitations, the 3s electron can be excited into states which have a principal quantum number n greater than or equal to four. The intermediate states are of the form 3sn,l,nhlh, where n,l, designates the orbital of the excited electron and n&,

Table 4. Sample

values

d

ofA, for f

Fe

At-

MO

___a-__

__-a___

___a_-_

(X 3s212f

1.65+09

5.01+09

1.31+10

3.08+10

(X 3s3p3d (X 3slp3d (X 3s3p4d

5.73+09 2.09+08

__-a___ ___a___ _-_a_--

---a___

2.95+08

___a-__

7.57+09 9.73+09

___a--_

___a___

1.20+08 2.95+08 9.91+06 5.52+06

5.14+09 X+:: 1:43:08

__-a___ __-a___ __-a___ ___a___

___b___ ___b___

4.89+10 7.02+10

4.09+11 1.12+12

3.98+12

___tJ___ ___b_____b___

3.03+10 2.69+10 3.43+09

3.17+11 4.07+11 1.08~11

3.49+12 5.87+12 2.24+12

___b___ ___b___ ___b_--__b___ -__b___

3.03+10 1.36+10 2.27+09 6.77+08 2.66+07

3.17+11 2.01+11 6.34+10 2.48+10 6.38+08

2.85+12 1.2>+12 5.65+11 1.16+10

(x)3s3p4s

___b___ ___b___

3.03+10 7.88+09

4.03+11 4.02+11

___a___ ___a--_

(x)3s3P5d (x)3s3p4s (x)3s4s4p (X)3S4S4f

___b___ ___b___ ___b___ -__tJ__-

3.03+10 6.35+09 2.70+08 2.74+08

3.17+11 2.28+11 4.32+10

3.49+12 3.96+12 1.15+12 7.65+11

___b___ ___b___ ___b___ ___b___ ___b___

4.17+09 1.80+08 2.78+07 1.13+08 1.78+08

9.91+09 2.20+09 3.69+09

2.25+12 7.45+11 3.05+11 3.19+10 5.53+10

2.63+07

3.84+09 4.69+10 1.04+1a

9.22+10 3.97+11 8.33+10

---a-__ ___a___ ---a___

3s3p5d 3s3dQp

(X 3s3d4f (X (x

(X 3S3P5f

3s3d4s 3s4s4d

(X

3s3p6f

(X (X

3s3d4s 3s4s4d 3s4s5d 3s4s5g

(X (X

(x)3s3p4d

(x)3s4s4p

are the same

3.39+08

(X (X

(x)3s4s6d

Si

Notations

(X 3s3p3d

(X 3s3p3d

(x)3s4s6f

the ions S?, Ar6’, Fe14+, and Mo3”. as in Table 3.

5.71+09 6.29+08

5.31+09 1.34+11 3.18+10

___a_-_

---a___ ---a___ ---a___

1.67+13

3.49+12

M. P. DUBS

18 Table 5. Values of 1 aUR(i,1,-

P( (I/.

i = 3x2, k,,7’ = Il.76 Ry: d) for the ion Si*'for 3s. An = 0 excitations;

Lah~'Sull coupling and cascade correction:ratesin unitsof IO " cm'/sec.

i

L

ab ”

-3

3s3pn(3)

(n b

4)

a.47

4

4

2

2

3s3pn(4)

5 3

5 3

6.41 1.06

3s3pn(Z)

3 1

3 1

5.12 0.561

3s3pn(l)

2 0

2 0

4.33 0.557

3s3pn(5)

6 4

6 4

3.91 0.402

3s3pnC6)

I 5

7 5

1.92 0.0415

3s3pnCO)

1

1

1.31

3s3pn(7)

8 6

8 6

0.362 0.0021

1.66

3s3dnC2)

0.0810 0.0478 0.0090

3s3dnCl)

O.OOOti4 o.0002

3s3dn(O)

O.O?'/"

Table 6. Values of 2 c?(i,

I, -

for 35, An = 0 excitations; I = 3.j‘.k,,'I' = 37.3 Ry;

d) for the ion A?'

LUh:'Sohcoupling and cascade correction;ratesin unitsof IO " cm'/sec.

-

i 2

3s

d=3sn,Panb(!b)

I

L

DR(i,jC

Gbd

ab

\

5 3

7.46 0.583

6 4

6 4

6.82 1.19

3s3pnC3)

4 2

4 i

6.69 2.28

3s3pn(6)

7 5

'I 5

5.55 0.749

3s3pn(7)

a 6

:

4.03 0.290

3S3P3Cl)

3

3 1

2.96 0.586

3s3pn(8)

3

7

9 7

2.55 0.0546

3s3pnC2)

2 0

2 0

1.78 1.58

3s3pn(O)

1

1

1.49

3s3pnC4)

5 3

3s3pnC5)

1

3s3dn(3)

5 3 1

5 3 1

0.428 0.285 0.118

3s3dn(2)

4 2 0

4 2 0

0.0685 0.0352 0.0059

3s3dn(l)

3

3 1

0.149 O.i)633

1

3s3dn(O;

2

2

-

i

I

0.076'5

d)

Dielectronic recombination rates for ions of the magnesium sequence Table 7. Values of c aDR(i, 1, -

d) for the ion Fe14+for

3s,

An = 0

excitations: i = 3s*,

19 kBT = 73.5

Ry;

Lo,,, S,h coupling and cascade corrections; rates in units of IO-” cm3/sec.

-

4

i

c -

L

3s3pnC5)

6 4

ab

cd

DR(i,&

4

t

5.73 1.26

7 5

5.59 1.01

z

',I9 0.775

:

5.02 0.697

9 7

11.18 0.342

4 2

4 2

3.86 2.00

10

10 a

3.28 0.116

11 9

2.26 0.0293

2 0

1.83 0.493

3 1

0.395 1.38

1

1.03

:

8

d)

"b

-

Table 8. Values of C aDR(i, I<-

d) for the ion MO”‘+for 3s, An = 0 excitations; i = 3x2, ksT = 206.0

Ry; L,:, Sy,, coupling and cascade corrections; rates in units of lOm’3cm3/sec.

i 7

3s

L

ab

3s3pnC7)

1

GbdDR(ipjc4

3s3pnC6)

2.75 0.450

2.69 0.559

3s3pnC5)

2.61 0.316

9 7

3s3pnC8)

2.56 0.298

I:

3s3pnC4)

5 3

3s3pnC9)

10 8

10

11

11

9

9

1.96 __-__

4 2

4 2

1.87 1.36

12 10

12 10

1.59

3 1

3 1

0.935

3s3pnCl)

2 0

2 0

0.988 0.278

3s3pnCO)

1

1

3s3pnClO)

3s3pnC3)

3s3pnCll)

3s3pnC2)

-

2.37 0.589 2.26 0.162

8

1.09

-

0.574

d)

20

M. P. Table 9. Values of 1 (UDR(i,I, angular

momentum

averaged

d) for the ion A?’ (AMA)

results

-

2

3s

Ed al.

for 3.5, &z # 0 excitations;

including cm’lsec

I 3s4snC3)

-

(n >, 4)

cascade

corrections:

I = 3.r’, k,,7’ = 32.3 Ry: rates

in units

I

& 3

CdDR(i,& +

I-

3s4snCl)

1

3s4prl'CO)

1

0.218

3s4snCO)

0

0.119

3s4pn(3)

4

0.0746

3s4dn'CO)

2

0.0572

3s‘Idn(3)

3

0.0272

3s4fn'CO)

0.407 0.272

3

0.0186

3s4dnC3)

5

0.0172

3s4fnC3)

4

0.0171

3s4sn'(4)

4

0.014'7

3s4fnC3)

5

-

d)

0.744

2

5)

IO ”

"b

3s4snC2)

(n'>,

of

-

-

I

i

DUBE

-I

0.0112

designates the orbital of the captured electron or vice versa. In a typical series of calculations, n, (of which 4 is the dominant case), I,, and lh are held fixed, and nh is varied from a minimum value (typically = 4) to infinity. Since there are a very large number of intermediate states to consider in this case, the angular-momentum-averaged (AMA) approximation was used in the calculation of aUR for all the intermediate states of the Are’ ion. From this set, a subset of dominant states is obtained, which contributes to 70-80% of the total rate, and the DR rate coefficients

Table

IO. Values of c n”R(i. Ry: L,,,. .I$,,,coupling

/( -

d) for the ion Ar”

and including

cascade

I = 3s’. X,,T

for 31, Zw + 0 excitations:

corrections:

rates in units of IO ” cm’/sec

-

4

i

L

s ab

ab

CdDR(i,l,

d

"b

2

3s4pn'(O)

(n', 5

3s4pnC3)

1 0

0.367 0.148 0.515

1 0

0.169 0.09$1 0.2641

1 0

0.251 +g

-

3s4snC2)

1 0

3s4snCl)

1 0

0.120 0.0625 0.1825

3s4sn(0)

1 0

0.0194 0.0582 0.0776

3s4dn'CO)

1 0

-

-

0.0323

dj

31.3

Dielectronic recombination rates for ions of the magnesium sequence Table II. Values of c olDR(i,I, -

21

d) for the ion S?’ for 3s, An f 0 excitations; i = 3s’. kBT = 11.76

Ry; Lo,,, .SIh coupling including cascade corrections; rates in units of IO-” cm-‘/set.

- -

i

L

1, 3s4pn’ (0)

3s4sn(1)

(n’>,5 1

ab

s ab

1

0.000647 0.000162 0.000809

1

(n&4)

2

3s4p4(3)

0.0000720 0.0000002 0.0000722

4

;-

-

-

are recalculated using the full LS coupling scheme. This same dominant set is then used for the remaining ions, with the ratio of CDR(AMA) for the dominant set to CDR(AMA) for the total set calculated for the Ar ion assumed to be the same for all the ions. The minimum value of the principal quantum number nh = no can be determined Table 12. Values of C tiDR(i, I,. -

d) for the ion Fe14+for 3s, An # 0 excitations: i = 3s*, k,,T = 73.5

Ry; Lo,,. So,, coupling and including cascade corrections: rates in units of IO-” cm’/sec.

i 3s2

_I___ I,

3s4snC3)

(II>, 4)

3s4sn(2)

Lab

3

3

2

3s4sn(1)

2

1

1

sab 1 0

1 0

1 0

2.60 1.99 -4.59 1.38

0.873 -=Fz 1.10 0.555 1.655

3s4sn'(4)

(n'35

4

4

1 0

0.941

3s4dn"(5)

(n"36

7

7

1 0

0.487 0.250

3s4pn’ (0)

1

1

1 0

3s4pn(3)

4

4

0.737 0.506 0.228 0.734

3s4dn'(4)

6

6

il 1

0.620 -%g 0.298

0 -s%3s4dn'CO)

2 :,

1 0

3

0

-

1 0

-I

0.395 0.186 0.581 0.211

-it+% 0.109 0.228 0.337

M. P. DUR~ PI N/.

12

from an expression similar to Eq. (7). Again, explicit calculations for C? are done only for the first several values of n,,, with the remaining values of aDR obtained by scaling. This was discussed in Part A of this section in connection with the 3s, An = 0 excitations. The rate coefficients for the most important transitions for the ion Ar” are given in Table 9 in the (AMA) approximation. In Tables lo- 13. the rate coefficients (yDRare given for the dominant transitions in the LS coupling scheme for the 3s, An # 0 excitations for the ions Si’+, Ar6+, Fe14+, and MO’“‘.

The variation of arDR(i, I,) the ions Si’+, A#‘+, Fe14’, and depicted for each of the four k,]Tf2. k[,T, 2kBT, and 3kljT.

vs 1, for the 3s, An = 0 excitations is shown in Fig. I for Mo3”‘. In Fig. 2. the variation of CU”~(~)vs temperature is ions. Explicit calculations were performed at energies of where knT is the scaled temperature for each ion, with

k,T - Z’. In Fig. 3, cyDR is plotted vs the nuclear charge &. For each ion. ~y”~ was computed at the scaled temperature, and the contributions made by the 3s. An # 0 and 3s. An = 0 excitations appear separately. 4. All the computations _L,,I,J’UI, coupling scheme,

lahle

ALTERNATE

COIJPLING

described in Sec. 3 for the DR rate coefficients where u and h denote the quantum numbers

13. Values of C
Ry; I.,,,., S,,, coupling

d) for the ion MO”” for 33, _ln f 0 excitations:

and includmg

-

cascade

corrections;

IPC t

L

I

-----s 3s

were made in the of the two active

1 = 3s:. XHT ~z X6.0

rates in units of IO

” cm’/scr

-

-

i

ab

--

I

s ab 1 0

6.30 5.71 X-G

2

1 0

2.91 1.98 4.89

) 5 (n'2,5:

6

1 0

2.06 2.04 -xx-c

3s4dn"(5)

(n"36‘1 1

7

1 iI

7.21 1.38 3.59

3s4sn'(4)

4

4

1 0

2.52 0.988 3.508

3s4fn'CO)

3

3

1 II

0.494 2.07 2.564

3s4sn(l)

1

1

1 0

1.64 0.793 z-m-

3s4dn(3)

5

5

1 0

0.283

3s4dn'CO)

2

2

1 0

7

1 0

1.14 0.553 1.693

1 0

0.512 0.925 1.437

3

3

2

3s4dn'(4)

t 3s4snC2)

-

PROCEDURES

3s4fn'(4)

7

3s4dnC3)

3 -

I

3

-

-k% I.13 %%

Dielectronic recombination rates for ions of the magnesium sequence

I IO -12 _

I

,

4 , ”

.

0 .

’

. 0

10“3dL

,

I

I

1

a

0

.

0 0

0

A

:

, Ry

32.3

Ry

A Fe:

73.5Ry

o MO:

206

I

I

11.8

Ry

)I

0 0”

,”

I Si

0 Ar

b

. 0

, l

0

.

.

I

:

23

0

.

. 0 0

0

.

0

0

Fig. 1. Values of aDR(i, I,) vs I
excited electrons in the intermediate state, denoted by (x)3sn,l,n~lh. In the three electron coupling scheme, the a and b electrons are coupled first to form L&Q,. Then, this pair is coupled to the spectator electron, which in this case is the 3s electron. Therefore, the intermediate states in the La&h coupling scheme use 1(x)3s, l,l~[L,fi,~,], LS). The formulae for the Auger decay rates in this LS coupling scheme are given in the Appendix. However, when (n,l,) is in the same shell as 3.5 it is more appropriate to couple them first. We consider for example an intermediate state given by 3s3p8d. The spectator

I

I

1.0

1.5 x

I

2.0

I

I

2.5

3.0

kBT

Fig. 2. Values of aDR(i) vs T (temperature); i = ls22s22p”3s*; all 3s

(- - -), Fe14+ (-. _. _. -) and Mo30+(-).

excitations;

Si’+

(. . . . .), Ar6’

24 11f-

/

12 -

/

/

/

//

/

/ An%0

I I

13_

SI

kgT:11.8

: :

Ar Fe MO:

Ry

kgT: 32.3 Ry kgTr73.5

Ry

$T=206

Ry

IL _

15

,

15

I

I

I

,

I

20

25

30

35

LO

ZC

Fig. 3.

values of (~?i)

vs

ZC, for ?G, An f 0 (dashed curve): 3.5, In = 0 (solid curve): computed temperature.

at scaled

electron in 3s should be coupled to the n,l, = 3p electron first, and then this resultant is coupled to the n& = 8d. Denoting the spectator electron by n,/, = 3s, this is called the Lo,&, coupling scheme. Thus, we have the recoupled state I(x)l,/,[L,,S,,]. h,LS). A threeelectron recoupling formula’ is used to recouple the system from the former case to the latter, and the correct formulae are given in the Appendix. coupling scheme Several sample calculations for uDR were performed using the LJ,, and the results were cnmpared with that obtained previously using the Lu&t, coupling scheme. Evidently, individual (YDR for each term state are changed drastically. But when summed over L,, and Srrr, little or no change in the DR rate coefficients were found in

Table

14. Values of A, for various

ions in the Lufiuh and the L,,S,, coupling SCJhor S,,, .

42

d

2 2

3S3P6f

4 4

3s3p6d

1 1

3~3~6~

2 2

-

s denotes

-

IOtl

schemes.

I -

4.18+13 1.76+13 5.94+13

2.37+13 3.57+13 5.94+13

2.13+14 1.24+14

1.46+14 1.91+14

3.37+14

3.37+14

4.10+13 3.03+13 7.13+13

7.13+13

1.16+14 8.73+13 2.03+14

9.45+13 1.09+14 2.03+14

either

Dielectronic Table

recombination

rates for ions of the magnesium

15. Values of A, for various

IOIl

ions in the LuJOh and the L,,&, coupling

d

I.

at

25

sequence schemes.

ArW

I 3s3p6f

Arc+

3s26f

0 1

Mo30+3s3plZf

I

6.16+10

1

0.0

the examples that were checked. This result can be understood of course in terms of the unitarity of the recoupling matrix. The results of these calculations are given below in Tables 14-16. In Table 14, the Auger decay rates are given for several sample cases using the both coupling schemes. In each example there is a redistribution of the singlet and triplet values of A,, but their sum remains unchanged. In Table 15, there is also a change in the radiative decay rates. However, Table 16 for the DR rate coefficients shows that the total rate remains nearly unchanged. A slight change in the aDR occurred for the ion Mo3’+, which has a larger value of the fluorescence yield w(d) than the other ions. Thus, when w(d) is very small, - gJ, and, therefore any unitarity transformation will leave the total aDR w(d) 4 1, CYDR invariant, so long as the fluorescence yields do not change drastically for the different intermediate states of the group being considered. This will also be true in the other extreme case of w(d) = 1. 5.

SUMMARY

We have presented the DR rate coefficients (YDR for the ions of the Mg isoelectronic sequence, in which only the 3s electron excitations is considered. The contribution of the 2s, 2p, and 1s excitations are not treated here, since it is important only at higher temperatures. The results presented here should be especially useful in the study of low temperature plasmas of light ions. Throughout the calculation, we have adopted the single-configuration, nonrelativistic approximation in LS coupling. Alternate coupling schemes (Log& vs L,,S,,) were compared, with minimal variations resulted in the total aDR. Extensions of the present work to include the effect of configuration mixing, intermediate coupling, relativistic corrections, and the contribution of the 2s, 2p, and 1s excitations will be presented elsewhere. Table

16. Values of (~“~(i, I, -

- -

10n

Ar

6+

Ar6+

M030+

M030+

d) for various ions in the Lo&h and LJ,, either S,, or S,,.

Id

I

2

1

2

0

2.04 4.97

;

1 0

1.17 0.64 1.81

2 2

1 0

5.15 4.69 TmT

3s3pbd

3S3P12f

3s4s4f

1 0

z

-

-

schemes.

-

4 -

3s3pbf

coupling

-

2.93

2.19 --w

1.60 -kg

0.37 1.44

1.81

4.63 -M

s denotes

M. P. DUBE CI d.

26 AcknoMdudKc~menr-This

work was supported

in part hy a DOE grant

REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. IO.

1I.

C. Breton, C. DeMichelis. M. Finkenthal. and M. Mattioli, Phy,s. Rev Lett. 41, 110 (1977). TFR Group, Plasma Phys. 19, 587 (1977). A. Burgess, Astro~hJa. J. 139, 776 (1964). A. Burgess, Astrophy.,-. J. 141, 1588 (1965). K. J. LaGattuta and Y. Hahn. Phj:v Rev. A24, 785 (198 I). Y. Hahn. Comments in .4f. M&c Phw. 13. 103 (1983). J. N. Gau and Y. Hahn. JQSRT23, 121 (1980). K. J. LaGattuta and Y. Hahn, Ph,rs. Lrrr. 84A, 468 (198 I). E. U. Condon and H. Odahasi. In Alomit, Strucrm The Universtty Press. Cambridge (1980). Y. Hahn. J. N. Gau. R. Luddy, M. Duhe. and N. Shkolnik. JQSRT 24, 505 (1980). C. E. Moore. Selected Tables of Atomic Spectra. jZ’SRIIS-h’BS3. National Bureau of Standards, DC 20025 (1975).

Washington.

APPENDIX

L,, vs Lab couplings

ofthe

uuger \tidth A,

The Auger width ,&(d - i, 1) will he defined by two different the explicit LS coupling scheme. The L,,, coupling scheme gives d f ((n,l,)“‘S,L,,(n,L)(n,,L,)S,,,.L,,,,. SL; If (n,/,)““’ is a closed shell. for example have

.4( LSL,/S‘,,,) =

where R,,,, = 2 for a = h and =

21,+ I F

<,i,

couplings

of the three outer shell electrons

i - I(n,/,)“‘+‘S;L;. k,/,

3s’. then S; = 1.1 = 0. Retaining

(Al)

. SL)

the S,,,,L,,,SL quantum

6.\,,d,1,(21,,+ 1)(21,, + 1)(2.%,,, + I)(ZL,,,, + I)I(S ,,,,L,,,$.

I for 0 # 11.The quantity

in

numbers.

we

(A2)

I is given as

(43)

Averaged

over the set of quantum

numbers

LSL,,,,S,,,, we have the AMA result

A, = 2 c C(?S

I d\“bl

with & = C(2S I\

+

(44)

I )(2L + I ,.~.(L.SL,,,,S‘,,.)IX,,,

\

+ l)(ZL + I). Thus. for (I f h 21,+1 .‘I,, = ~ 8

(A51

c z QS,,, + I)(21 -,,,, + I)I(S ,,,,L.,,$. i “Cbh

and for ~1= h ‘4 = (2/< + 101” +

0

4(41,, +

1)

I) x CS”~,+ I )(2S”h+ I)&S”/&,)?

(A6)

I u&h

with S,,, + L,,,, = even. (Here S,,, = S’,, and L,,,, = L,,,.) In the L,, coupling scheme. we have. again with i?? ~= I.

d - l(n,l,)(n,l,)S,,L,,,,(n,,l,,&&. SL) - 1- i(n,l,,*s;L;,/c,I,. SL). In Eq. (A7), a closed shell with S; + L; = 0 is again assumed. recoupling formula was used and the result is given by A,(LSL,,S,,)

= c c (2L,,, + I u&h

In this case a three-electron

1)(ZS<,,. + 1)(2L.,,+ I WL + 1) x (I:, :’

2fl/2 ?:I;.)I l,2

l/2

(.A7) angular

momentum

s,,, )

,,2 St<,]..l‘,(LSI~,,I~~~,,,) (AX)

wtth ,4,(LSL,,,S,,b) given by Eq. (A2). .Averaging over the final set of quantum numbers LSL,,,&,, wc recover the same results given by Eqs. (A5) and (A6). The correct formulae for the radiative decay rates .il, are given elsewhere,“’ and will not he repeated here. The angular momentum recoupling of the three-electron system does not introduce any new formulae for 1,.